IF `1^3+2^3+…….+9^3=2025` then the value of `(0.11)^3 +(0.22)^3+……+(0.99)^3` is close to

Options

A. 0.2695
B. 2.695
C. 3.695
D. 0.3695

Answer

To solve the problem, we need to find the value of \( (0.11)^3 + (0.22)^3 + \ldots + (0.99)^3 \) given that \( 1^3 + 2^3 + \ldots + 9^3 = 2025 \).

### Step-by-Step Solution:

1. **Understanding the Series**:
The series we need to evaluate can be rewritten in terms of fractions:
\[ (0.11)^3 + (0.22)^3 + (0.33)^3 + \ldots + (0.99)^3 = \left(\frac{11}{100}\right)^3 + \left(\frac{22}{100}\right)^3 + \ldots + \left(\frac{99}{100}\right)^3 \]

2. **Factoring Out Common Terms**:
Notice that each term can be expressed as:
\[ \left(\frac{n \cdot 11}{100}\right)^3 \quad \text{for } n = 1, 2, \ldots, 9 \]
Thus, we can factor out \( \left(\frac{11}{100}\right)^3 \):
\[ \left(\frac{11}{100}\right)^3 \left(1^3 + 2^3 + \ldots + 9^3\right) \]

3. **Substituting the Known Value**:
We know from the problem statement that:
\[ 1^3 + 2^3 + \ldots + 9^3 = 2025 \]
Therefore, substituting this into our expression gives:
\[ \left(\frac{11}{100}\right)^3 \cdot 2025 \]

4. **Calculating \( \left(\frac{11}{100}\right)^3 \)**:
First, calculate \( 11^3 \):
\[ 11^3 = 1331 \]
Then, calculate \( 100^3 \):
\[ 100^3 = 1000000 \]
So, we have:
\[ \left(\frac{11}{100}\right)^3 = \frac{1331}{1000000} \]

5. **Final Calculation**:
Now, we can combine everything:
\[ \frac{1331}{1000000} \cdot 2025 = \frac{1331 \cdot 2025}{1000000} \]

6. **Calculating the Product**:
Calculate \( 1331 \cdot 2025 \):
\[ 1331 \cdot 2025 = 2694975 \]
Therefore:
\[ \frac{2694975}{1000000} = 2.694975 \]

7. **Approximating the Result**:
The approximate value of \( 2.694975 \) can be rounded to \( 2.69 \).

### Conclusion:
Thus, the value of \( (0.11)^3 + (0.22)^3 + \ldots + (0.99)^3 \) is close to **2.69**.